Enumeration of a class of sequences generated by inversions

Any permutation of a finite field F-q can be represented by a polynomial P-n(x) = (. . . ((a(0)x a(1))(q-2) + a(2))(q-2) ... a(n))(q-2) + a(n+1), for some n >= 0. In this note we present the number of distinct permutations of the types P-2(x) and P-3(x) with full cycle. These results extend earlier work on the inversive pseudorandom number generator and on P-1.